A216962 Triangle read by rows, arising in enumeration of permutations by cyclic peaks.

1, 2, 5, 1, 15, 9, 52, 63, 5, 203, 416, 101, 877, 2741, 1361, 61, 4140, 18425, 15602, 2153, 21147, 127603, 165786, 46959, 1385, 115975, 914508, 1700220, 823256, 74841, 678570, 6794508, 17200589, 12802659, 2389953, 50521, 4213597, 52354116, 173871641, 185499899, 59207070, 3855277

Offset: 1

Author

N. J. A. Sloane, Sep 27 2012

Keywords

• nonn
• tabf

Comments

See Ma and Chow (2012) for precise definition (see corollary 2).

Examples

			Triangle begins:
1
2
5,1
15,9
52,63,5
203,416,101
877,2741,1361,61
...
		

Links

• Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264, 2012

Crossrefs

First column is A000110.

Programs

Mathematica

P[1] := x y; P[n_] := P[n] = ((n-1)q + x y) P[n-1] + 2q(1-q) D[P[n-1], q] + x(1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify;
row[n_] := CoefficientList[P[n] /. {x -> 1, y -> 1}, q];
Array[row, 12] // Flatten (* Jean-François Alcover, Sep 25 2018 *)

PARI

tabf(m) = {P = x; for (n=1, m, M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = (n*q+x)*P + 2*q*(1-q)*deriv(P, q)+ x*(1-q)*deriv(P,x););} \\ Michel Marcus, Feb 08 2013

Extensions

More terms from Michel Marcus, Feb 08 2013